Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. 365–375.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
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STRONG SOLUTIONS FOR THE NAVIER-STOKES EQUATIONS
ON BOUNDED AND UNBOUNDED DOMAINS WITH A
MOVING BOUNDARY
JÜRGEN SAAL
Abstract. It is proved under mild regularity assumptions on the data that
the Navier-Stokes equations in bounded and unbounded noncylindrical regions
admit a unique local-in-time strong solution. The result is based on maximal
regularity estimates for the in spatial regions with a moving boundary obtained
in [16] and the contraction mapping principle.
1. Introduction and main result
S
For T > 0 let QT := t∈(0,T ) Ω(t) × {t} ⊆ Rn+1 be a noncylindrical space-time
domain. In this note we consider the Navier-Stokes equations
vt − ∆v + (v · ∇)v + ∇p = f in QT ,
div v = 0
v=0
in QT ,
on ∪t∈(0,T ) ∂Ω(t) × {t},
v|t=0 = v0
(1.1)
in Ω(0) =: Ω0 ,
with velocity field v and pressure p. Here we assume the moving boundary, i.e. the
evolution of the domain Ω(t) to be determined by the level-preserving diffeomorphism
ψ : Ω0 × (0, T ) → QT , (ξ, t) 7→ (x, t) = ψ(ξ, t) := (φ(ξ; t), t)
such that for each t ∈ [0, T ), φ(·; t) maps Ω0 onto Ω(t). More precisely we assume
the following conditions on φ respectively ψ.
assumption 1.1. Let T ∈ (0, ∞), Ω0 ⊆ Rn be a domain of class C 3 either bounded,
exterior, or a perturbed half-space. Suppose that the domains Ω(t), t ∈ [0, T ], are
all of the same type as Ω0 , i.e. {Ω(t)}t∈[0,T ] is either a family of bounded domains,
a family of exterior domains, or a family of perturbed half-space s. Furthermore:
(1) For each t ∈ [0, T ], φ(·; t) : Ω0 → Ω(t) is a C 3 -diffeomorphism. Its inverse
we denote by φ−1 (·; t) (to emphasize that φ−1 is merely the inverse w.r.t.
the space variables we use the semicolon notation (ξ; t) for the argument of
φ and φ−1 ).
2000 Mathematics Subject Classification. 35Q30, 76D05.
Key words and phrases. Navier-Stokes equations; moving boundary; maximal regularity.
c
2007
Texas State University - San Marcos.
Published February 28, 2007.
365
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J. SAAL
EJDE/CONF/15
(2) For φ regarded as a function from Q0T := Ω0 × (0, T ) into Rn we assume
φ ∈ Cb3,1 (Q0T ) := {f ∈ C(Q0T ) : ∂tk Dxα f ∈ Cb (Q0T ), 1 ≤ 2k + |α| ≤ 3, k ∈
N0 , α ∈ Nn0 }, where Cb (Q0T ) denotes the space of all bounded and continuous
functions on Q0T .
(3) We have det ∇ξ φ(ξ, t) ≡ 1, (ξ, t) ∈ Q0T , (volume preserving).
Let us remark that in view of realistic physical situations problem
(1.1) should
S
be considered with a certain boundary condition v = b 6= 0 at t∈(0,T ) ∂Ω(t) × {t}.
On the other hand, by assuming the existence of a solenoidal field β such that
β = b, the problem with b 6= 0 can be reduced to the case b = 0 as described in
[9] and [8]. Therefore we restrict our considerations to the system (1.1) with zero
boundary conditions.
Also, note that in certain concrete situations the existence of the diffeormorpism
ψ is established. For instance in [8] the authors give as a nice example of a moving
domain Ω(t) a bowl with swimming goldfishes (note that kisses are not allowed).
The existence of ψ in such a situation is proved in [12] and [5].
Now define I p (A) := (X, D(A))1− p1 ,p , for 1 < p < ∞, where the latter space
denotes the real interpolation space of a Banach space X and the domain D(A) of
a closed operator A in X. For t ∈ [0, T ) we denote by
AΩ(t) = −PΩ(t) ∆
D(AΩ(t) ) = W
2,q
(Ω(t)) ∩
defined on
W01,q (Ω(t))
∩ Lqσ (Ω(t))
as usual the Stokes operator in the space of solenoidal fields
Lq (Ω(t))
∞ (Ω(t))
Lqσ (Ω(t)) = Cc,σ
,
∞
where Cc,σ
(Ω(t)) := {u ∈ Cc∞ (Ω(t)) : div u = 0}. Here PΩ(t) : Lq (Ω(t)) →
q
Lσ (Ω(t)) denotes the Helmholtz projection associated to the Helmholtz decompoc 1,q (Ω(t))}.
sition Lq (Ω(t)) = Lqσ (Ω(t)) ⊕ Gq (Ω(t)), where Gq (Ω(t)) = {∇p : p ∈ W
Note that the existence of a compatible family {PΩ,q }q∈(1,∞) of bounded projections
PΩ = PΩ,q : Lq (Ω) → Lqσ (Ω) is well known for all types of domains Ω considered
in this note, see e.g. [7, 21, 19]. For the above types of moving domains our main
result is as follows.
Theorem 1.2. Let n ≥ 2, (n + 2)/3 < q < ∞, and T ∈ (0, ∞]. Let the evolution of
Ω(t), t ∈ [0, T ], be determined by a function ψ satisfying Assumptions 1.1. Then,
for each v0 ∈ I p (AΩ0 ) and f ∈ Lp ((0, T ); Lq (Ω(t))) there exists a T ∗ ∈ (0, T ) and
a unique solution (v, p) of problem (1.1), such that
v ∈ W 1,q ((0, T ∗ ); Lq (Ω(t))) ∩ Lq ((0, T ∗ ); D(AΩ(t) )),
c 1,q (Ω(t))).
p ∈ Lq ((0, T ∗ ); W
Remark 1.3. By an inspection of the proof one will realize that the assumption on
the family {Ω(t)}t∈[0,T ] , which we intrinsically use, is not the particular geometric
shape of the domains, but the property of having maximal regularity of the corresponding Stokes operator AΩ(t) for each fixed t ∈ [0, T ]. Thus, Theorem 2.1 stays
true for each family {Ω(t)}t∈[0,T ] with Ω(t) of the “same type” for all t ∈ [0, T ] such
that AΩ(t) has maximal regularity. This property for instance is also known to be
valid for families of asymptotically flat layers as examined in [2, 1].
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Some special cases of the situation in Theorem 2.1 are considered in several
former works. First investigations of the solvability of (1.1) can be found in [18].
However, until now there are only existence results available under the restrictive
assumptions that q = 2, i.e. in the Hilbert space case, and that {Ω(t)}t∈[0,T ] is a
family of bounded domains. In that situation for instance in [8] the existence of
a unique local-in-time solution of (1.1) is proved. The existence of global weak
solutions in L2 (Ω(t)) for the Navier-Stokes equations was already shown in [6] (see
also [3]). The periodic case was obtained in [11], i.e. if t 7→ Ω(t) is periodic then
there exists a periodic weak solution of the Navier-Stokes equations . A result
for more regular periodic solutions can be found in [10]. Another existence result
of local-in-time strong solutions of the Navier-Stokes equations in L2σ (Ω(t)) for
bounded Ω(t) is obtained in [14]. There the authors could relax a restrictive decay
condition on the right hand side f assumed in [8], but nevertheless they were able
to construct a more regular solution.
We want to remark that the assumptions on the evolution and regularity of Ω(t)
differ in the above cited papers. This depends mainly on the method that the
authors use in their works. The approach presented here is closely related to the
method used in [8]. Therefore we have similar assumptions on ψ (hence also on
Ω(t)) as in [8].
Theorem 2.1 generalizes the above cited results on (1.1) in several directions.
Firstly, we handle Lq -spaces for the full scale (n + 2)/3 < q < ∞ and arbitrary
space dimension n ≥ 2, and not only the Hilbert space case if n = 2, 3. Moreover,
we prove the existence of strong solutions under mild regularity assumptions on the
data. Secondly, our approach also covers various families {Ω(t)}t∈[0,T ) of unbounded
domains.
As we already mentioned, Theorem 2.1 is based on maximal regularity estimates
for the corresponding linear Stokes equations obtained in [16]. Therefore, in Section 2 we recall the main results given in [16] in a slightly adapted form suitable
for our purposes. Utilizing the maximal regularity for the Stokes equations and the
contraction mapping principle in Section 3 we give the proof of our main result,
the unique local-in-time strong solutions to the (1.1) on noncylindrical space-time
domains.
We introduce some notation used in the sequel. By C k (Ω) we denote the space
of all k-times continuously differentiable functions in an open subset Ω of Rn , and
by Cbk (Ω) its subspace of k-times bounded continuously differentiable functions.
Pk
As usual W k,q (Ω) is the Sobolev space with norm k · kk,q = ( j=0 k∇j · kqq )1/q
and Lq (Ω) = W 0,q (Ω) the Lebesgue space of q-integrable functions. We also make
c 1,q (Ω), consisting of all locally integrable
use of the homogeneous Sobolev space W
functions f in Ω with k∇f kq < ∞, modulo constants. Note that we do not distinguish between scalar and vector valued Sobolev spaces, i.e. we write Lq (Ω) for
(Lq (Ω))n , W k,q (Ω) for (W k,q (Ω))n , etc. Furthermore, L(X, Y ) denotes the class of
all bounded operators from X to Y and Isom(X, Y ) its subclass of isomorphisms.
If X = Y we set L(X) := L(X, X) and Isom(X) := Isom(X, X). The domain of
an operator A in a Banach space X we denote by D(A), its range by R(A), and its
resolvent set by ρ(A).
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2. Maximal regularity for the Stokes equations
For the readers convenience we recall here the basic steps that lead to the maximal regularity result for the linearized version of (1.1) obtained in [16]. Also note
that we state them in a slightly adapted form as we will need it in Section 3. In this
context we restrict the statements here to the case of finite T > 0, but note that
under suitable additional assumptions all the assertions are still true for T = ∞.
Employing the notation of the last section, here we are concerned with the linear
problem
vt − ∆v + +∇p = f in QT ,
div v = 0 in QT ,
v = 0 on ∪t∈(0,T ) ∂Ω(t) × {t},
v|t=0 = v0
(2.1)
in Ω(0) =: Ω0 .
For this system, in [16, Theorem 2.1], the following result is proved.
Theorem 2.1. Let n ≥ 2, 1 < q < ∞, and T ∈ (0, ∞). Let the evolution of Ω(t),
t ∈ [0, T ], be determined by a function ψ satisfying Assumptions 1.1. Then problem
c 1,q (Ω(t)), t ∈ [0, T ].
(2.1) has a unique solution t 7→ (v(t), p(t)) ∈ D(AΩ(t) ) × W
Furthermore, this solution satisfies the estimate
Z Th
i
kvt (t)kpLq (Ω(t)) + kv(t)kpW 2,q (Ω(t)) + k∇p(t)kpLq (Ω(t)) dt
0
(2.2)
Z T
≤ C(T ) kv0 kpI p (AΩ
0)
+
0
kf (t)kpLq (Ω(t)) dt
for all v0 ∈ I p (AΩ0 ) and f ∈ Lp ((0, T ); Lq (Ω(t))). If v0 = 0, then the constant
C(T ) in (2.2) is uniformly bounded from above on finite intervals, more precisely
there is a T0 > 0 and a C(T0 ) > 0 such that C(T ) ≤ C(T0 ) for all T ≤ T0 .
The proof of this result relies on a transform of (2.1) via ψ to a problem on the
cylindrical domain Ω0 × (0, T ). The price we have to pay is that we are then left
with a nonautonomous system of partial differential equations, i.e. the coefficients
of these transformed equations depend on space and time in general. Here Assumption 1.1 (2) assures that they are at least continuous. Another important point is
that the transformed functions belong to the solenoidal space Lqσ (Ω0 ), which relies
essentially on Assumption 1.1 (3). More precisely this condition assures that the
operator div is invariant under the chosen transform.
Similar to the autonomous Stokes equations this will give us the possibility to
formulate an associated abstract Cauchy problem with operators acting in Lqσ (Ω0 ).
The idea here is to use the family of projections PΩ0 (t) : Lq (Ω0 ) → Lqσ (Ω0 ), which
are exactly the transformed Helmholtz projections PΩ(t) .
First let us list some obvious consequences of Assumption 1.1. In view of
det ∇φ(ξ, t) ≡ 1 and ψ(ξ, t) = (φ(ξ; t), t) we also have det ∇ψ = 1. Moreover, Assumption 1.1 (2) implies ψ ∈ Cb3,1 (Q0T ; Rn+1 ). In virtue of the implicit function theorem we therefore have ψ −1 ∈ Cb3,1 (QT ; Rn+1 ) and since ψ −1 (x, t) = (φ−1 (x; t), t),
(x, t) ∈ QT , also φ−1 ∈ Cb3,1 (QT ; Rn ).
We transform (2.1) to a system on a fixed domain as follows. For a function
v : QT → Cn set
ṽ(ξ, t) := v(φ(ξ; t), t),
(ξ, t) ∈ Ω0 × [0, T ].
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Then
(∇x v)(φ(ξ; t), t) = (∇ξ φ)−T ∇ξ ṽ (ξ, t),
where M
−T
T −1
denotes (M )
(2.3)
T
and M stands for the transposed Matrix. Now define
u(ξ, t) := (Φ(t)v)(ξ, t) := (∇ξ φ)−1 ṽ (ξ, t),
Assumption 1.1 (1), (2), and (3) on φ imply that
(ξ, t) ∈ Ω0 × [0, T ].
(2.4)
Φ(t) ∈ Isom(W k,q (Ω(t)), W k,q (Ω0 )) ∩ Isom(W0k,q (Ω(t)), W0k,q (Ω0 ))
for k = 0, 1, 2 and t ∈ [0, T ], and we even have the uniform estimates
kΦ(t)vkW k,p (Ω0 ) ≤ C1 kvkW k,p (Ω(t)) ≤ C2 kΦ(t)vkW k,p (Ω0 )
(2.5)
k,p
for all v ∈ W (Ω(t)), t ∈ [0, T ], k = 0, 1, 2. It is also easy to see that ν(x, t) is the
outer normal at ∂Ω(t) in x if and only if µ(ξ, t) = (∇φ)T (ξ, t)ν(φ(ξ, t)) is the outer
normal at ∂Ω0 in ξ. This implies ν · v = 0 if and only if µ · Φv = 0. Furthermore,
under Assumption 1.1 (in particular (3)) in [8, Proposition 2.4]1 it is proved that
divξ u(ξ, t) = divx v(φ(ξ; t), t),
Lqσ (Ω(t))
(ξ, t) ∈ Ω0 × [0, T ].
Lqσ (Ω0 )
is an isomorphism as well. This
→
This implies that Φ(t) :
property of Φ, which is essential in what follows, is the reason why we have to
choose the special transform given in (2.4). On the other hand note that this
transform is responsible for the fact, that we have to assume C 3 boundary instead
of C 2 only.
In view of (2.3) it is clear that Φ(t)∆x Φ(t)−1 has a representation as
X
Φ(t)∆x Φ(t)−1 =
aα (·, t)Dα
(2.6)
|α|≤2
|α|
with
|α|,
certain matrices aα ∈ Cb 2 (Ω × (0, T ))n×n , |α| ≤ 2. Explicitly
(ξ, t) = [(∇ξ φ)−1 (∇ξ φ)−T ∇ξ · (∇ξ φ)−T ∇ξ (∇ξ φ)u](ξ, t)
n
X
=
we have
(∂xk φ−1 )(∂xj φ−1 )i (∂xj φ−1 )` (φ(ξ; t), t)
i,j,k,`,m=1
h
(2.7)
k
m
k
× (∂ξ` ∂ξi ∂ξm φ )u + (∂ξi ∂ξm φ )∂ξ` u
m
i
+ (∂ξ` ∂ξm φk )∂ξi um + (∂ξm φk )∂ξ` ∂ξi um (ξ, t).
We also have
∂t v(x, t) = ∂t [(∇ξ φ)u](φ−1 (x; t), t)
=
n
X
(∂t φ−1 )j (x; t) (∂ξi ∂ξj φ)ui + (∂ξi φ)∂ξj ui (φ−1 (x; t), t)
i,j=1
n
X
+
(2.8)
(∂ξi ∂t φ)ui + (∂ξi φ)∂t ui (φ−1 (x; t), t).
i=1
Thus
Φ(t)∂t Φ(t)−1 = ∂t +
X
bβ (·, t)Dβ
(2.9)
|β|≤1
1Actually in [8] only bounded Ω are treated. But since it is a pointwise condition the proof
0
given there applies to each Ω ⊂ Rn .
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2|β|,|β|
with certain bβ ∈ Cb
(Ω × (0, T ))n×n , |β| ≤ 1. If we set F := Φf and
φ
u0 := Φ(0)v0 , as well as ∇ (t) := (∇ξ φ(t))−1 (∇ξ φ(t))−T ∇ξ and p̃ := p ◦ ψ, the
transformed equations on Q0T = Ω × (0, T ) become
X
X
ut +
bβ D β u −
aα Dα u + ∇φ (·)p̃ = F in Q0T ,
|β|≤1
|α|≤2
in Q0T ,
div u = 0
u=0
(2.10)
on ∂Ω0 × (0, T ),
u|t=0 = u0
in Ω0 .
Since Φ(t) is an isomorphism, clearly (u, p̃) satisfies (2.10) if and only if (v, p) fulfills
(2.1). Obviously
PΩ0 (t) := Φ(t)PΩ(t) Φ(t)−1 : Lq (Ω0 ) → Lqσ (Ω0 ),
t ∈ [0, T ],
is again a projection, where PΩ(t) denotes the Helmholtz projection on Lq (Ω(t)).
Note that
c 1,q (Ω(t))}.
Gq (t) := (I − PΩ0 (t))Lq (Ω0 ) = {∇φ (t)(π ◦ ψ); π ∈ W
Thus, PΩ0 (t) is not the Helmholtz projection on Lq (Ω0 ) in general. As Gq (t)
depends on t we see that also the projection PΩ0 (t) does, although its range Lqσ (Ω0 )
is independent of t. Defining
X
AΩ0 (t) := −PΩ0 (t)
aα (·, t)Dα
(2.11)
|α|≤2
on
D(AΩ0 (t)) = Φ(t)D(AΩ(t) )
= W 2,q (Ω0 ) ∩ W01,q (Ω0 ) ∩ Lqσ (Ω0 )
= D(AΩ0 ),
t ∈ [0, T ],
and
B(t) := PΩ0 (t)
X
bβ (·, t)Dβ ,
t ∈ [0, T ],
(2.12)
|β|≤1
the system (2.10) can be rewritten as the nonautonomous Cauchy problem
u0 (t) + (AΩ0 (t) + B(t))u(t) = F (t),
u(0) = u0 ,
t ∈ (0, T ),
(2.13)
on the space Lqσ (Ω0 ). Observe that AΩ0 (t) = Φ(t)AΩ(t) Φ(t)−1 , i.e. it is exactly
the transformed Stokes operator on Ω(t) for t ∈ [0, T ]. Moreover, we see that
the domain of AΩ0 (t) does not depend on t and equals the domain of the Stokes
operator AΩ0 in Lqσ (Ω0 ).
For T ∈ (0, ∞) and p ∈ (1, ∞) we denote by MRp (X, K) the class of all operators
(and propagators) A(·) having maximal (Lp -) regularity on X with a maximal
regularity constant not exceeding K, i.e. there exists a unique solution t 7→ u(t) ∈
D(A(t)) of the (eventually nonautonomous) Cauchy problem
u0 + A(·)u = f,
in (0, T ),
u(0) = u0 ,
(2.14)
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satisfying the estimate
ku0 kW 1,p ((0,T );X) + kA(·)ukLp ((0,T );X) ≤ K(kf kLp ((0,T );X) + ku0 kI p (A(0)) )
for f ∈ Lp ((0, T ); X) and u0 ∈ I p (A(0)).
Based on two abstract results for nonautonomous systems (see [16, Teorem 1.4
and Theorem 2.5]) the following result is obtained in [16, Theorem 3.5].
Proposition 2.2. Let T ∈ (0, ∞). Let Ω0 , φ be as in Assumption 1.1 and the
families {AΩ0 (t)}t∈[0,T ] and {B(t)}t∈[0,T ] be defined as in (2.11) and (2.12), respectively. Then for µ > 0 large enough we have
k(µ + AΩ0 (t) + B(t))(µ + AΩ0 (s) + B(s))−1 kL(X) ≤ C,
and AΩ0 (·) + B(·) ∈
t, s ∈ [0, T ).
(2.15)
MR(Lqσ (Ω0 ), C(T )).
We turn to the proof of the maximal regularity result for (2.1).
Proof of Theorem 2.1. Observe that in view of (2.15) and the equivalence of the
norms k · k2,q and k · kD(AΩ0 (0)+B(0)) we have
Z T
Z T
ku0 (t)kpq + ku(t)kp2,q dt ≤ C(T )
kF (t)kpq dt + ku0 kpI p .
(2.16)
0
0
This yields
Z
T
k(∂t +
0
X
bβ (t))u(t)kpq + ku(t)kp2,q + k∇φ (t)p̃(t)kpq dt
|β|≤1
Z
≤ C(T )
0
T
kF (t)kpq dt + ku0 kpI p .
for the solution (u, p) of (2.10). In view of (2.5), (2.9), and since {Φ(t)}t∈[0,T ] is a
family of isomorphisms, this implies estimate (2.2) for the solution of the original
Ω(t)
equations (SE)f,v0 . If v0 = 0 and f ∈ Lq ((0, T ); Lqσ (Ω(t))) we may extent f trivial
to the interval (0, T0 ), where we denote the extended function by f¯. Let (u, p) and
Ω(t)
(ū, p̄) be the solution to problem (2.1) and (SE)f¯,0 , respectively. The uniqueness
of the solution implies (ū, p̄)|(0,T ) = (u, p). By this fact it easily follows that the
constants C(T ) in (2.2) can be dominated by a constant C(T0 ) for all T ≤ T0 . This
completes the proof.
3. Strong solutions for the Navier-Stokes equations
Utilizing the maximal regularity for the Stokes equations, in this section we prove
our main result Theorem 1.2. In order to estimate the nonlinear term in (1.1), a
further main ingredient in the proof will be the following embedding.
Lemma 3.1. Let T > 0, J = (0, T ), a ≥ 2, and q >
n
a
+ 1. Then we have
W 1,q (J; Lq (Ω(t))) ∩ Lq (J; W 2,q (Ω(t))) ,→ L2q (J; W 1,aq/(a−1) (Ω(t))).
If we replace W 1,q (J; Lq (Ω(t))) by W01,q (J; Lq (Ω(t))) on the left hand side, then
there exists a T0 > 0 such that the embedding constant is governed by a constant
C(T0 ) > 0 for all T ≤ T0 .
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Proof. Note that (2.5) and Assumption 1.1 (2) imply that
Φ ∈Isom(W `,p (J; W k,q (Ω(t))), W `,p (J; W k,q (Ω0 )))
∩ Isom(W0`,p (J; W k,q (Ω(t))), W0`,p (J; W k,q (Ω0 )))
for ` = 0, 1, k = 0, 1, 2, and 1 ≤ p, q ≤ ∞. In particular we have
kΦvkW `,p (J;W k,q (Ω0 )) ≤ C1 kvkW `,p (J;W k,q (Ω(t))) ≤ C2 kΦvkW `,p (J;W k,q (Ω0 ))
`,p
(3.1)
k,q
for all v ∈ W (J; W (Ω(t))), ` = 0, 1, k = 0, 1, 2, with C1 , C2 independent of
T > 0. Therefore it suffices to prove the embedding
W 1,q (J; Lq (Ω0 )) ∩ Lq (J; W 2,q (Ω0 )) ,→ L2q (J; W 1,aq/(a−1) (Ω0 )),
and that this embedding is even valid with an embedding constant independent of
T ≤ T0 , if we assume zero time trace at t = 0.
It is a known fact that for s ∈ [0, 1],
W 1,q (J; Lq (Ω0 )) ∩ Lq (J; W 2,q (Ω0 )) ,→ W s,q (J; W 2(1−s),q (Ω0 )).
(3.2)
This follows e.g. by an application of the mixed derivative theorem [20] (see also
[15]) for J = R and Ω0 = Rn . Employing suitable extension operators in space and
time it can be seen that this embedding is still valid for J = (0, T ) and our Ω0 , even
with an embedding constant independent of T ≤ T0 , if we assume vanishing time
trace at t = 0 (see e.g. [15, Proposition 6.1] for the existence of such an extension
operator). According to q > na −1 we can find an > 0 such that q > na −1+. Now
set s := (1 + )/2q. Since 1 − sq > 0 and 2q < q/(1 − sq) the Sobolev embedding
implies
kvkL2q (J;W 2(1−s),q (Ω0 )) ≤ CkvkW s,q (J;W 2(1−s),q (Ω0 )) .
Furthermore, we have n − sq > 0 and aq/(a − 1) < nq/(n − (1 − 2s)). Thus, we
may apply the Sobolev embedding also in space to the result
kvkL2q (J;W aq/(a−1),q (Ω0 )) ≤ CkvkL2q (J;W 2(1−s),q (Ω0 ))
≤ CkvkW s,q (J;W 2(1−s),q (Ω0 ))
s,q
2(1−s),q
for v ∈ W (J; W
(Ω0 )). In combination with (3.1) and (3.2) this yields the
first assertion. The additional assertion follows by the fact that also the embedding
constant of the Sobolev embedding in time can be chosen independently of T ≤ T0 ,
if we assume vanishing time trace at t = 0.
Finally we prove our main result by employing the contraction mapping principle.
Proof of Theorem 1.2. First let us introduce some notation. We set
E = E1 × E2 ,
F = F1 × F2
with
E1 := W 1,q ((0, T ); Lq (Ω(t))) ∩ Lq ((0, T ); D(AΩ(t) )),
c 1,q (Ω(t))),
E2 := Lq ((0, T ); W
F1 := Lp ((0, T ); Lq (Ω(t))),
F2 := I p (AΩ0 ).
Also, denote by E0 and F0 the corresponding spaces with vanishing time trace at t =
0, that is E0 = E1,0 × E2 with E1,0 := W01,q ((0, T ); Lq (Ω(t))) ∩ Lq ((0, T ); D(AΩ(t) ))
and F0 := F1 × {0}. Now let LT be the solution operator of the linear problem
EJDE-2007/CONF/15
STRONG SOLUTIONS
373
Ω(t)
(SE)·,· and observe that according to Theorem 2.1 we have LT ∈ Isom(E, F).
Then problem (1.1) can formally be rephrased as
LT (v, p) = (f + F (v), v0 ),
(3.3)
where F (v) := −(v · ∇)v. For further purposes it will be convenient to split off the
part corresponding to the data f and v0 . To do so let (v ∗ , p∗ ) be the solution to
Ω(t)
(SE)f,v0 , i.e.
(v ∗ , p∗ ) = L−1
T (f, v0 ).
Moreover, we set v̄ := v − v ∗ and p̄ = p − p∗ . By this notation (3.3) turns into
LT (v̄, p̄) = (f + F (v̄ + v ∗ ), v0 ) − LT (v ∗ , p∗ )
= (F (v̄ + v ∗ ), 0) =: H0 (v̄, p̄),
and therefore the fixed point equation reads as
(v̄, p̄) = L−1
T H0 (v̄, p̄),
where v̄ and H0 (v̄, p̄) now have zero time trace by construction. Next suppose a ≥ 2
and that
n
q > + 1.
(3.4)
a
Then Lemma 3.1 implies E1 ,→ L2q (J, Laq/(a−1) (Ω(t))). For u, w ∈ E1 we therefore
deduce by applying first the Hölder and then the Sobolev inequality (in space)
k(u · ∇)wkF1 ≤ kukL2q (J,Laq (Ω(t))) k∇wkL2q (J,Laq/(a−1) (Ω(t)))
≤ CkukL2q (J,W 1,aq/(a−1) (Ω(t))) kwkL2q (J,W 1,aq/(a−1) (Ω(t))) .
(3.5)
Observe that the above application of the Sobolev inequality requires a second
condition on q, namely that
a−2
.
(3.6)
q>n
a
Since relation (3.4) is decreasing in a and (3.6) is increasing in a, the best possible
value for q is reached at the intersection point of the graphs of the two equations
y = na + 1 and y = n a−2
a , which is
3n
, (n + 2)/3 .
(a, y) =
n−1
3n
Thus, by the assumption q > (n + 2)/3 and by setting a = n−1
the two conditions
(3.4) and (3.6) are satisfied, which justifies the application of Lemma 3.1 and the
Sobolev embedding in estimate (3.5).
Now fix T0 > 0. Let Br (0) ⊆ E0 be the ball around 0 with radius r, and
(v̄, p̄) ∈ Br (0). Applying (3.5) to H0 (v̄, p̄) yields
kH0 (v̄, p̄)kF ≤ kF (v̄ + v ∗ )kF1
≤ C kv̄k2a,q + kv̄ka,q kv ∗ ka,q + kv ∗ k2a,q ,
where k · ka,q denotes the norm of the space L2q (J; W 1,aq/(a−1) (Ω(t))). Applying
Lemma 3.1 to the terms involving v̄ results in
kH0 (v̄, p̄)kF ≤ C k(v̄, p̄)k2E0 + k(v̄, p̄)kE0 kv ∗ ka,q + kv ∗ k2a,q
(3.7)
374
J. SAAL
EJDE/CONF/15
for all T ≤ T0 in view of (v̄, p̄) ∈ E0 . Note that by definition H0 ∈ L(E0 , F0 ).
According to Theorem 2.1 we have kL−1
T kL(F0 ,E0 ) ≤ C(T0 ) for all T ≤ T0 . Hence,
there exists a constant C0 > 0 independent of T ≤ T0 such that
−1
kL−1
T H0 (v̄, p̄)kE0 ≤ kLT kL(F0 ,E0 ) kH0 (v̄, p̄)kF
≤ C0 k(v̄, p̄)k2E0 + k(v̄, p̄)kE0 kv ∗ ka,q + kv ∗ k2a,q .
Observe that v ∗ is a fixed function only depending on the data (f, v0 ). Hence we
may choose r > 0 small so that r < max{1, 1/3C0 } and then T > 0 small such that
r
.
kv ∗ ka,q <
3C0
This implies that
kL−1
T H0 (v̄, p̄)kE0 ≤ r,
−1
that is L−1
T H0 (Br (0)) ⊆ Br (0). To see that LT H0 is a contraction observe that
−1
kL−1
T H0 (v̄1 , p̄1 ) − LT H0 (v̄2 , p̄2 )kE0
≤ kL−1
T kL(F0 ,E0 ) kH0 (v̄, p̄) − H0 (v̄, p̄)kF0
≤ C(T0 ) k[(v̄1 − v̄2 ) · ∇]v ∗ kF1 + k(v ∗ · ∇)(v̄1 − v̄2 )kF1
+ k[(v̄1 − v̄2 ) · ∇]v̄1 kF1 + k(v̄2 · ∇)(v̄1 − v̄2 )kF1 .
By applying (3.5) and Lemma 3.1 we obtain in a similar way as above that
−1
kL−1
T H0 (v̄1 , p̄1 ) − LT H0 (v̄2 , p̄2 )kE0
≤ C0 kv ∗ ka,q + k(v̄1 , p̄1 )kE0 + k(v̄2 , p̄2 )kE0 k(v̄1 − v̄2 , p̄1 − p̄2 )kE0
with a constant C0 > 0 not depending on T ≤ T0 . Consequently, if we choose
T, r > 0 such that r, kv ∗ ka,q < 1/(6C0 ), we see that L−1
T H0 : Br (0) → Br (0) is a
contraction and the assertion follows by the contraction mapping principle.
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Jürgen Saal
Department of Mathematics and Statistics, University of Konstanz, Box D 187, 78457
Konstanz, Germany
E-mail address: juergen.saal@uni-konstanz.de